class: center, middle, inverse, title-slide .title[ # Teoria da Produção ] .subtitle[ ## Apêndice - Curto e Longo Prazos ] .author[ ### Prof. Mauricio Uriona Maldonado ] .institute[ ### EPS 5222 ] .date[ ### 29/09/2022 ] --- class: center, middle # Produção no Curto Prazo --- # Função de Produção no curto prazo - Vamos começar entendendo a forma de uma função de produção quando apenas um insumo é variável. | L| Q| |--:|-----:| | 1| 4.5| | 2| 18.0| | 4| 60.0| | 5| 82.5| | 6| 102.0| | 7| 115.5| | 8| 120.0| | 9| 112.5| | 10| 90.0| --- # Graficar - Os dados da tabela podem ser representados como uma função de produção: `$$Q_{(L)}=-0.5L^3+6L^2-L$$` -- <img src="index_files/figure-html/unnamed-chunk-2-1.png" width="576" /> --- # Calculando PMg e PMe - O PMg pode ser entendido como a derivada `$$\frac{\partial Q_{(L)}}{\partial L}$$` -- - No nosso caso: `$$\frac{\partial Q_{(L)}}{\partial L}=-1.5L^2+12L-1$$` -- - Já, o PMe pode ser entendido como o produto médio por unidade de trabalho: `$$\frac{Q_{(L)}}{L}$$` -- - No nosso caso: `$$\frac{Q_{(L)}}{L}=-0.5L^2+6L-1$$` --- # Tabela com Q, L, PMg e PMe <div id="htmlwidget-3bf504a73bd0e84548be" style="width:100%;height:auto;" class="datatables html-widget"></div> <script type="application/json" 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class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>L<\/th>\n <th>Q<\/th>\n <th>PMg<\/th>\n <th>PMe<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"columnDefs":[{"className":"dt-right","targets":[1,2,3,4]},{"orderable":false,"targets":0}],"order":[],"autoWidth":false,"orderClasses":false}},"evals":[],"jsHooks":[]}</script> --- # Graficando o PMg e o PMe (1 de 2) -- - Vamos graficar o PMg e o PMe para entender suas trajetórias -- <img src="index_files/figure-html/unnamed-chunk-4-1.png" width="576" style="display: block; margin: auto;" /> -- - Conforme a figura, o PMg cresce até atingir um máximo e logo cair rapidamente, já o PMe apresenta um comportamento mais estável. --- # Graficando o PMg e o PMe (2 de 2) - Uma propriedade interessante é que o máximo do PMe é exatamente o ponto onde cruza com o PMg. -- -- <img src="index_files/figure-html/unnamed-chunk-5-1.png" width="576" style="display: block; margin: auto;" /> --- class: center, middle, inverse # Produção no Longo Prazo --- # Função de Produção no Longo Prazo -- - No longo prazo, a função de produção `\(Q\)` depende do Trabalho `\(L\)` e do Capital `\(K\)` -- - Uma função de produção comumente usada é a *Cobb-Douglas* que tem a forma: `$$Q_{(K,L)}=AK^\alpha L^\beta$$` -- - Onde: `\(A\)` é uma constante e `\(\alpha+\beta=1\)` -- - Vamos assumir que `\(\alpha=0.5\)` e que `\(\beta=0.5\)` --- # Isoquantas <div id="htmlwidget-c403718eccd39bb1fa39" style="width:576px;height:360px;" class="plotly html-widget"></div> <script type="application/json" 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--- # Rendimentos de escala -- - Rendimentos crescentes: quando a produção mais do que dobra se dobrarem os insumos -- - Rendimentos constantes: quando a produção dobra proporcionalmente a dobrarem os insumos -- - Rendimentos decrescentes: quando a produção cresce menos do que o dobro se dobrarem os insumos -- - Em funções Cobb-Douglas, é fácil identificar o tipo de rendimentos à escala -- - Se `\(\alpha+\beta < 1\)` então **Rendimentos Decrescentes** -- - Se `\(\alpha+\beta = 1\)` então **Rendimentos Constantes** -- - Se `\(\alpha+\beta > 1\)` então **Rendimentos Crescentes** --- # Rendimentos de escala constantes ```r ggplotly(crs) ``` <div id="htmlwidget-16171950682a9c38078a" style="width:504px;height:504px;" class="plotly html-widget"></div> <script type="application/json" 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u800"],"type":"scatter","mode":"lines","line":{"width":1.88976377952756,"color":"rgba(199,124,255,1)","dash":"solid"},"hoveron":"points","name":"u800","legendgroup":"u800","showlegend":true,"xaxis":"x","yaxis":"y","hoverinfo":"text","frame":null}],"layout":{"margin":{"t":23.3059360730594,"r":7.30593607305936,"b":53.4661685346617,"l":85.3466168534662},"plot_bgcolor":"rgba(255,255,255,1)","paper_bgcolor":"rgba(255,255,255,1)","font":{"color":"rgba(0,0,0,1)","family":"","size":14.6118721461187},"xaxis":{"domain":[0,1],"automargin":true,"type":"linear","autorange":false,"range":[0,210],"tickmode":"array","ticktext":["0","20","40","60","80","100","120","140","160","180","200"],"tickvals":[0,20,40,60,80,100,120,140,160,180,200],"categoryorder":"array","categoryarray":["0","20","40","60","80","100","120","140","160","180","200"],"nticks":null,"ticks":"outside","tickcolor":"rgba(51,51,51,1)","ticklen":3.65296803652968,"tickwidth":0.66417600664176,"showticklabels":true,"tickfont":{"color":"rgba(77,77,77,1)","family":"","size":21.2536322125363},"tickangle":-0,"showline":true,"linecolor":"rgba(0,0,0,1)","linewidth":0.66417600664176,"showgrid":false,"gridcolor":null,"gridwidth":0,"zeroline":false,"anchor":"y","title":{"text":"<b> Trabalho (L) <\/b>","font":{"color":"rgba(0,0,0,1)","family":"","size":21.2536322125363}},"hoverformat":".2f"},"yaxis":{"domain":[0,1],"automargin":true,"type":"linear","autorange":false,"range":[0,8100],"tickmode":"array","ticktext":["0","1,000","2,000","3,000","4,000","5,000","6,000","7,000","8,000"],"tickvals":[0,1000,2000,3000,4000,5000,6000,7000,8000],"categoryorder":"array","categoryarray":["0","1,000","2,000","3,000","4,000","5,000","6,000","7,000","8,000"],"nticks":null,"ticks":"outside","tickcolor":"rgba(51,51,51,1)","ticklen":3.65296803652968,"tickwidth":0.66417600664176,"showticklabels":true,"tickfont":{"color":"rgba(77,77,77,1)","family":"","size":21.2536322125363},"tickangle":-0,"showline":true,"linecolor":"rgba(0,0,0,1)","linewidth":0.66417600664176,"showgrid":false,"gridcolor":null,"gridwidth":0,"zeroline":false,"anchor":"x","title":{"text":"<b> Capital (K) <\/b>","font":{"color":"rgba(0,0,0,1)","family":"","size":21.2536322125363}},"hoverformat":".2f"},"shapes":[{"type":"rect","fillcolor":null,"line":{"color":null,"width":0,"linetype":[]},"yref":"paper","xref":"paper","x0":0,"x1":1,"y0":0,"y1":1}],"showlegend":false,"legend":{"bgcolor":"rgba(255,255,255,1)","bordercolor":"transparent","borderwidth":1.88976377952756,"font":{"color":"rgba(0,0,0,1)","family":"","size":11.689497716895}},"hovermode":"closest","barmode":"relative"},"config":{"doubleClick":"reset","modeBarButtonsToAdd":["hoverclosest","hovercompare"],"showSendToCloud":false},"source":"A","attrs":{"6c817e32cb8c":{"x":{},"y":{},"colour":{},"type":"scatter"}},"cur_data":"6c817e32cb8c","visdat":{"6c817e32cb8c":["function (y) ","x"]},"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.2,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script> --- # Rendimentos de escala crescentes - `\(\alpha\)` = 0.6 e `\(\beta\)` = 1 ```r utility_Ki <- function(my_L, my_Q) { (my_Q^(1/0.6))/my_L^(1/0.6) } irs <- rs %>% mutate(u100 = utility_Ki(x, 100), u200 = utility_Ki(x,200), u400 = utility_Ki(x,400), u800 = utility_Ki(x,800)) %>% pivot_longer(2:5, values_to = "K") %>% ggplot(aes(x, K, color=name))+ geom_line()+ scale_x_continuous(expand = c(0, 0), breaks = seq(0, 200, 20), labels = scales::comma) + scale_y_continuous(expand = c(0, 0), breaks = seq(0, 8000, 1000), labels = scales::comma) + coord_cartesian(xlim = c(0, 210), ylim = c(0, 8100))+ theme_classic()+ theme(axis.text=element_text(size=16), axis.title=element_text(size=16, face="bold"), legend.position = "none")+ labs(x = "Trabalho (L)", y = "Capital (K)") ggplotly(irs) ``` <div id="htmlwidget-e643efce304daf56c33a" style="width:504px;height:504px;" class="plotly html-widget"></div> <script type="application/json" 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Trabalho (L) <\/b>","font":{"color":"rgba(0,0,0,1)","family":"","size":21.2536322125363}},"hoverformat":".2f"},"yaxis":{"domain":[0,1],"automargin":true,"type":"linear","autorange":false,"range":[0,8100],"tickmode":"array","ticktext":["0","1,000","2,000","3,000","4,000","5,000","6,000","7,000","8,000"],"tickvals":[0,1000,2000,3000,4000,5000,6000,7000,8000],"categoryorder":"array","categoryarray":["0","1,000","2,000","3,000","4,000","5,000","6,000","7,000","8,000"],"nticks":null,"ticks":"outside","tickcolor":"rgba(51,51,51,1)","ticklen":3.65296803652968,"tickwidth":0.66417600664176,"showticklabels":true,"tickfont":{"color":"rgba(77,77,77,1)","family":"","size":21.2536322125363},"tickangle":-0,"showline":true,"linecolor":"rgba(0,0,0,1)","linewidth":0.66417600664176,"showgrid":false,"gridcolor":null,"gridwidth":0,"zeroline":false,"anchor":"x","title":{"text":"<b> Capital (K) <\/b>","font":{"color":"rgba(0,0,0,1)","family":"","size":21.2536322125363}},"hoverformat":".2f"},"shapes":[{"type":"rect","fillcolor":null,"line":{"color":null,"width":0,"linetype":[]},"yref":"paper","xref":"paper","x0":0,"x1":1,"y0":0,"y1":1}],"showlegend":false,"legend":{"bgcolor":"rgba(255,255,255,1)","bordercolor":"transparent","borderwidth":1.88976377952756,"font":{"color":"rgba(0,0,0,1)","family":"","size":11.689497716895}},"hovermode":"closest","barmode":"relative"},"config":{"doubleClick":"reset","modeBarButtonsToAdd":["hoverclosest","hovercompare"],"showSendToCloud":false},"source":"A","attrs":{"6c8171bd7905":{"x":{},"y":{},"colour":{},"type":"scatter"}},"cur_data":"6c8171bd7905","visdat":{"6c8171bd7905":["function (y) ","x"]},"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.2,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script> --- # Rendimentos de escala decrescentes - `\(\alpha\)` = 0.4 e `\(\beta\)` = 0.1 ```r utility_Kd <- function(my_L, my_Q) { (my_Q^(1/0.4))/my_L^(0.1/0.4) } drs <- rs %>% mutate(u100 = utility_Ki(x, 100), u200 = utility_Ki(x,200), u400 = utility_Ki(x,400), u800 = utility_Ki(x,800)) %>% pivot_longer(2:5, values_to = "K") %>% ggplot(aes(x, K, color=name))+ geom_line()+ scale_x_continuous(expand = c(0, 0), breaks = seq(0, 200, 20), labels = scales::comma) + scale_y_continuous(expand = c(0, 0), breaks = seq(0, 8000, 1000), labels = scales::comma) + coord_cartesian(xlim = c(0, 210), ylim = c(0, 8100))+ theme_classic()+ theme(axis.text=element_text(size=16), axis.title=element_text(size=16, face="bold"), legend.position = "none")+ labs(x = "Trabalho (L)", y = "Capital (K)") ggplotly(drs) ``` <div id="htmlwidget-7deb05fda1e2e7693e4e" style="width:504px;height:504px;" class="plotly html-widget"></div> <script type="application/json" 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